MINNEAPOLIS -- David Price said on Monday that he doesn't know where he'll be playing come July 31, the non-waiver Trade Deadline.

The talk at the American League media session leading up to Tuesday night's 85th All-Star Game at Target Field, touched on New York, Los Angeles, Chicago and even Tampa Bay, where the 28-year-old left-hander has played his entire seven-year big league career.

Given the choice, Price said he preferred to stay in his comfort zone and remain with the Rays. But if anyone in the Tampa Bay front office sought his opinion, did he have a choice of a new home?

"I mean, yeah, I probably would have a couple of teams I wouldn't mind going to," Price said. "I feel like I should probably keep those teams to myself."

Do those teams share any characteristics?

"They all play baseball," Price responded.

Would hitting be part of the equation, opening up a Pandora's Box to all the National League teams?

"I could hit for anybody," Price said.

And so the merry-go-round keeps twirling, with Price grasping toward the brass ring.

Price started for the Rays on Sunday at Tropicana Field and defeated the Blue Jays, 3-0, pitching eight-plus innings of five-hit, one-walk, five-strikeout ball. At the break, his record improved to 9-7 and his ERA dropped to 3.23.

Because of the Sunday start, Price exercised his option to sit out the Midsummer Classic. He was replaced on the AL roster by Fernando Rodney, a former closer for Tampa Bay, who's having a banner year with Seattle.

That didn't stop Price from taking a seat at his own personal podium and subjecting himself to numerous questions from the media. The Rays have him under control through 2015 and are in fourth place in the AL East, 9 1/2 games behind the division-leading Orioles. None of that may matter, and inquiring minds want to know Price's thoughts about the immediate future.

"Winning is absolutely something you want to do," Price said. "Being a part of something special is also something you want to do. You can take that to a first-place team, or you can take that to a last-place team like the Cubs. With the talent they have coming up, they could be a very special team in a few years as well. That would probably be the coolest city to win a championship in right now."

Price had his first chance with the Rays in 2008, when they went to the World Series and lost to the Phillies in five games. He was just a kid then. Price said he hasn't had concrete contract extension talks with Tampa Bay since '12, and none are on the horizon. And except for third baseman Evan Longoria, who the team signed to a pair of six-year extensions totaling $117.5 million, Price knows that the Rays shed star players like a lizard sheds its skin. Just ask Rodney, B.J. Upton, Matt Garza, Carlos Pena and James Shields, to name a few.

"No, no talks since 2012, the offseason going into 2013," said Price, who is earning $14 million this year on a one-year deal. "We both understood that in order for Tampa to continue the kind of success we've had over the past five or six years, this is the way they operate.

"I'd love to stay there and continue to be competitive, but I don't know if that's even a possibility. They signed Longoria, so it's not unheard of, but I don't know if they can do two people that way."

And so, the buzzards are hovering. With Jeff Samardzija and Jason Hammel already traded by the Cubs to the A's, Price is the premier pitcher left on the market. He is certainly right up there with left-hander Cliff Lee, the focal point of two non-waiver Deadline trades: in 2009 from the Indians to the Phillies, and in '10 from the Mariners to the Rangers.

Both times, Lee's new clubs went to the World Series. Both times, they lost: the Phils to the Yankees and the Rangers to the Giants. This year, Lee is on Philadelphia's disabled list with a sore left elbow and is not a factor in the market.

That makes Price a top commodity. The deep-pocketed Dodgers and Yankees could both use him, because of injuries to starting pitchers. Los Angeles has Josh Beckett on the DL with a sore hip, although his return may be imminent. New York has lost four-fifths of its rotation and is in dire need of some help if it intends to make up a five-game deficit in the division.

But the Dodgers have recently said they are not intent on giving up key Minor League prospects for Price, and the Yankees may not have them. In any event, despite what they've said publicly, the Rays may be loath to trade Price elsewhere in their own division.

About playing in a bigger market like New York or Los Angeles, Price said it might take an adjustment.

"It takes time to fit into a new clubhouse. I understand that," Price said. "But I think I have good people skills. I could be anywhere, actually. The Rays said they don't have a problem trading me in the division. That opens the doors for a couple of other teams. I don't think I'd have a problem fitting in anywhere. I know a lot of guys in the big leagues now. I at least know somebody from every team. That always helps.

"The Rays have traded quite a few guys away or let a few guys go who are in the big leagues now. I guess I'd just have to learn how to handle everything."